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Find the maximum profit under the continuous model.

Read the idea, work independently, then explain what changed.

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高一必修第一册(A版).pdf · 3.4 · PDF 100 / printed page 93

Revisit first: Power functions

TOPIC 01

Applications of functions I

Build understanding of applications of functions i through definitions, contrasting cases and justified applications.

What you will be able to explain

Interpret and use applications of functions i with domain checks.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Your turn

Find the maximum profit under the continuous model.

P(q)=-(q-7)^2+49,\quad q\ge0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Define the variable and its units, form a model, and check feasible inputs.

Hint 2

The squared term is nonnegative.

Worked solution

Define the variable and its units, form a model, and check feasible inputs.
Calculate or simplify this relation.
$P(q)\le49;\quad q=7$
The maximizing output is nonnegative and therefore feasible.

The requested value is 49.

Checks and common pitfalls: The maximizing output is nonnegative and therefore feasible.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Question sequence

What must be true before using the main rule for applications of functions i?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Board plan

Interpret and use applications of functions i with domain checks.
Modelling: State variables, units, assumptions and feasible inputs.
Validation: A fitted formula needs contextual checks and further evidence.
Close with: conditions → representation → reasoning → check.

Anticipated thinking

Expected reasoning: State variables, units, assumptions and feasible inputs.
Expected reasoning: A fitted formula needs contextual checks and further evidence.
Expected correction: Average speed is not generally the average of two speeds.

Assessment checklist

1: identify the givens and required quantity.
1: choose a valid definition, representation or method.
1: present connected, correct reasoning.
1: check conditions and explain the result.